Finding small regular graphs of girths 6, 8 and 12 as subgraphs of cages

Small kk-regular graphs of girth gg where g=6,8,12g=6,8,12 are obtained as subgraphs of minimal cages. More precisely, we obtain (k,6)(k,6)-graphs on 2(kq−1)2(kq−1) vertices, (k,8)(k,8)-graphs on 2k(q2−1)2k(q2−1) vertices and (k,12)(k,12)-graphs on 2kq2(q2−1)2kq2(q2−1), where qq is a prime power and kk is a positive integer such that q≥k≥3q≥k≥3. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth g=6,8,12g=6,8,12.